Open 3-Manifolds, Wild Subsets of S³ and Branched Coverings

In this paper, a representation of closed 3-manifolds as branched coverings of the 3-sphere, proved in Montesinos-Amilibia (2002), and showing a relationship between open 3-manifolds and wild knots and arcs will be illustrated by examples. It will be shown that there exist a 3-fold simple covering  p : S³ S³ branched over the remarkable simple closed curve of Fox (1949) (a wild knot). Moves are defined such that when applied to a branching set, the corresponding covering manifold remains unchanged, while the branching set changes and becomes wild. As a consequence every closed, oriented 3-manifold is represented as a 3-fold covering of S³ branched over a wild knot, in plenty of different ways, confirming the versatility of irregular branched coverings. Other collection of examples is obtained by pasting the members of an infinite sequence of two-component strongly-invertible link exteriors. These open 3-manifolds are shown to be 2-fold branched coverings of wild knots in the 3-sphere Two concrete examples, are studied: the solenoidal manifold, and the Whitehead manifold. Both  are 2-fold covering of the Euclidean space R³ branched over an uncountable collection of string projections in R³.

Key words: Wild knots, open manifolds, branched coverings.
2000 Mathematics Subject Classification: 57M12, 57M30, 57N10.